Simplify and expand the following expression: $ \dfrac{2}{n - 9}+ \dfrac{2}{n - 7}+ \dfrac{2}{n^2 - 16n + 63} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{2}{n^2 - 16n + 63} = \dfrac{2}{(n - 9)(n - 7)}$ Now we have: $ \dfrac{2}{n - 9}+ \dfrac{2}{n - 7}+ \dfrac{2}{(n - 9)(n - 7)} $ The least common multiple of the denominators is: $ (n - 9)(n - 7)$ In order to get the first term over $(n - 9)(n - 7)$ , multiply by $\dfrac{n - 7}{n - 7}$ $ \dfrac{2}{n - 9} \times \dfrac{n - 7}{n - 7} = \dfrac{2(n - 7)}{(n - 9)(n - 7)} $ In order to get the second term over $(n - 9)(n - 7)$ , multiply by $\dfrac{n - 9}{n - 9}$ $ \dfrac{2}{n - 7} \times \dfrac{n - 9}{n - 9} = \dfrac{2(n - 9)}{(n - 9)(n - 7)} $ Now we have: $ \dfrac{2(n - 7)}{(n - 9)(n - 7)} + \dfrac{2(n - 9)}{(n - 9)(n - 7)} + \dfrac{2}{(n - 9)(n - 7)} $ $ = \dfrac{ 2(n - 7) + 2(n - 9) + 2} {(n - 9)(n - 7)} $ Expand: $ = \dfrac{2n - 14 + 2n - 18 + 2}{n^2 - 16n + 63} $ $ = \dfrac{4n - 30}{n^2 - 16n + 63}$